The Slovenia-born 28-year-old broke the lucky loser twice in the first set and then twice in the second to secure a convincing triumph within 69 minutes in Hamburg.
Next up for world number 46 Bedene is Russia's Karen Khachanov, who overcame Germany's Rudolf Molleker 6-4 6-3 in the first round.
Veteran Spaniard David Ferrer overcame Hamburg debutant Nikoloz Basilashvili to progress.
Eighth seed Ferrer, a finalist in 2014, needed two hours to beat Georgian Basilashvili 6-2 3-6 6-3 and will next face Argentina's Federico Delbonis.
Jiri Vesely also had a three-set battle with Fernando Verdasco in prevailing 7-6 (9/7) 6-7 (0/7) 6-3.
Florian Mayer beat Andrey Kuznetsov 6-3 4-6 6-4 to set up a quarter-final meeting with sixth seed Diego Schwartzman, the Argentinian defeating Cedrik-Marcel Stebe 1-6 6-4 6-2.
At the Swiss Open in Gstaad, top seed David Goffin fended off four match points to win a tense encounter with Moldovan Radu Albot 2-6 7-6 (9/7) 7-6 (9/7).
Goffin, who reached the Gstaad final in 2015, was down 3/6 in the third-set tie-break.
But he reeled off three consecutive points and eventually crossed the finishing line to set up a quarter-final showdown with Dutchman Robin Haase.
The sixth-seeded Haase, a runner-up in 2013 and 2016, did not face a break point in a 7-5 7-5 win over Brazil's Thiago Monteiro.
Switzerland's Henri Laaksonen, Spaniard Roberto Bautista Agut and fifth seed Paolo Lorenzi all booked their places in the second round.
Second seed John Isner advanced to the quarter-finals of the BB&T Atlanta Open with a straight-sets victory over Vasek Pospisil of Canada.
Isner recorded a 6-3 6-4 victory to set up a last-eight clash with Lukas Lacko after he beat the sixth-seeded Donald Young 6-3 6-1.
Third seed Gilles Muller is also safely through to the quarter-finals following a 6-4 6-7 (1/7) 6-3 win over qualifier Quentin Halys.
In the final game of the day, American qualifier Tommy Paul beat Nalek Jaziri 6-7 (5/7) 6-4 6-3.
